Foundations — Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis

This page assumes you know nothing. Before we can even read "$y^2=4ax$", we have to earn every piece: what a coordinate is, what a point and a line look like on paper, what "distance" means for each, and why we are allowed to square both sides. We build them in order, each one leaning on the last.

1. The plane and a coordinate $(x,y)$

The picture. Draw two number-line rulers crossing at right angles. The horizontal one is the $x$-axis, the vertical one is the $y$-axis, and where they cross — the point $(0,0)$ — is the origin.

Why the topic needs it. A parabola is a set of points. To talk about "which points obey the walking rule" we first need a language for naming any point at all. That language is $(x,y)$. Later, $P=(x,y)$ will mean "a general point, we don't yet know where — let the rule decide".

2. A fixed point $F$ and a fixed line $\ell$

The picture. Just a dot and a straight line drawn near it, not touching.

Why "not passing through $F$"? If the line went through the dot, the walking rule would collapse (you could stand on both at once) — the curve would degenerate. We forbid it so a genuine curve exists. This is our first degenerate case flagged.

Notation note. The symbol $\ell$ is just a cursive letter "l" mathematicians use to name a line, exactly like we use $F$ to name a point. It carries no other meaning.

3. Distance from a point to a point — the ruler tool

Where does the square root come from? Draw a right triangle whose horizontal leg is the gap in the $x$-values, $x-a$, and whose vertical leg is the gap in the $y$-values, $y-0$. The straight-line distance $PF$ is the hypotenuse (the slanted long side). Pythagoras says ${\text{hyp}}^2={\text{leg}}^2+{\text{leg}}^2$, so

$P{F}^2=(x-a{)}^2+(y{)}^2⟹PF=\sqrt{(x-a{)}^2+y^2}.$

Why the squared terms are never negative. $(x-a{)}^2$ means a real number times itself; a number times itself is always $\ge 0$ regardless of sign. So the inside of the root is always $\ge 0$ and the root is always a real distance — no bad cases.

4. Distance from a point to a line — the perpendicular tool

The picture. From point $P$ drop a straight arrow straight onto the fence so it meets at $90^{\circ }$. That arrow's length is the distance. A crooked path forms a triangle whose slanted side is longer — so it loses.

The special easy case we actually use. When the line is vertical, say $x=-a$ (every point on it has $x$-coordinate $-a$), the perpendicular is horizontal. So the distance from $P=(x,y)$ to $x=-a$ is just the horizontal gap:

$\text{dist}(P,\ell )=|x-(-a)|=|x+a|.$

All cases covered. If $P$ is right of the fence, $x+a>0$ and $|x+a|=x+a$. If $P$ is left of it, $x+a<0$ and $|x+a|=-(x+a)$. If $P$ is on it, $x+a=0$, distance $0$. The bars handle all three without us splitting into cases by hand.

5. The walking rule as an equation

$\underset{\text{to focus }F}{\underbrace{\sqrt{(x-a{)}^2+y^2}}}=\underset{\text{to directrix }\ell }{\underbrace{|x+a|}}$

6. Why we are allowed to square both sides

$(x-a{)}^2+y^2=(x+a{)}^2.$

Expanding both brackets, $-2ax$ on the left and $+2ax$ on the right, the $x^2$ and $a^2$ cancel, leaving the parent's headline result $y^2=4ax$. You have now built it from nothing but a dot, a fence, and Pythagoras.

7. The coefficient written as $4a$ (a preview of why)

Prerequisite map

Equipment checklist

Connections

• Parent topic — Parabola standard forms
• Coordinate Geometry — Distance Formula (the ruler tool built here)
• Conic Sections — Overview
• Eccentricity and focus-directrix definition
• Ellipse — standard forms, Hyperbola — standard forms
• Tangents and Normals to a Parabola
• Projectile Motion